$f(x, y) = x^3 + x^2y + xy^2 + y^3$ What is $\nabla \cdot (\nabla f)$ ? $\nabla \cdot (\nabla f) = $
The Laplacian of a scalar field $f$ is the sum of each of its second partial derivatives. $\nabla \cdot (\nabla f) = \dfrac{\partial^2 f}{\partial x^2} + \dfrac{\partial^2 f}{\partial y^2}$ [What does it mean to take a dot product with the gradient?] Let's find the second partial derivatives of $f$ ! $\begin{aligned} f_{xx} &= \dfrac{\partial}{\partial x} \left[ \dfrac{\partial f}{\partial x} \right] \\ \\ &= \dfrac{\partial}{\partial x} \left[ 3x^2 + 2xy + y^2 \right] \\ \\ &= 6x + 2y \\ \\ f_{yy} &= \dfrac{\partial}{\partial y} \left[ \dfrac{\partial f}{\partial y} \right] \\ \\ &= \dfrac{\partial}{\partial y} \left[ x^2 + 2xy + 3y^2 \right] \\ \\ &= 2x + 6y \end{aligned}$ The Laplacian is $\nabla \cdot (\nabla f) = f_{xx} + f_{yy}$. Therefore: $\nabla \cdot (\nabla f) = 8x + 8y$